and,
fromList,
head,
+ ifoldl,
length,
map,
maximum,
import qualified Algebra.Transcendental as Transcendental ( C )
import qualified Prelude as P ( map )
+-- | Our main matrix type.
data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
+
+-- Type synonyms for n-by-n matrices.
type Mat1 a = Mat N1 N1 a
type Mat2 a = Mat N2 N2 a
type Mat3 a = Mat N3 N3 a
type Mat4 a = Mat N4 N4 a
type Mat5 a = Mat N5 N5 a
+-- | Type synonym for row vectors expressed as 1-by-n matrices.
+type Row n a = Mat N1 n a
+
+-- Type synonyms for 1-by-n row "vectors".
+type Row1 a = Row N1 a
+type Row2 a = Row N2 a
+type Row3 a = Row N3 a
+type Row4 a = Row N4 a
+type Row5 a = Row N5 a
+
+-- | Type synonym for column vectors expressed as n-by-1 matrices.
+type Col n a = Mat n N1 a
+
+-- Type synonyms for n-by-1 column "vectors".
+type Col1 a = Col N1 a
+type Col2 a = Col N2 a
+type Col3 a = Col N3 a
+type Col4 a = Col N4 a
+type Col5 a = Col N5 a
+
+-- We need a big column for Gaussian quadrature.
+type N10 = S (S (S (S (S N5))))
+type Col10 a = Col N10 a
+
+
instance (Eq a) => Eq (Mat m n a) where
-- | Compare a row at a time.
--
row (Mat rows) i = rows ! i
--- | Return the @j@th column of @m@. Unsafe.
-column :: Mat m n a -> Int -> (Vec m a)
-column (Mat rows) j =
- V.map (element j) rows
+-- | Return the @i@th row of @m@ as a matrix. Unsafe.
+row' :: (Arity m, Arity n) => Mat m n a -> Int -> Row n a
+row' m i =
+ construct lambda
where
- element = flip (!)
+ lambda _ j = m !!! (i, j)
+
+
+-- | Return the @j@th column of @m@. Unsafe.
+--column :: Mat m n a -> Int -> (Vec m a)
+--column (Mat rows) j =
+-- V.map (element j) rows
+-- where
+-- element = flip (!)
+-- | Return the @j@th column of @m@ as a matrix. Unsafe.
+column :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
+column m j =
+ construct lambda
+ where
+ lambda i _ = m !!! (i, j)
-- | Transpose @m@; switch it's columns and its rows. This is a dirty
--- implementation.. it would be a little cleaner to use imap, but it
--- doesn't seem to work.
+-- implementation, but I don't see a better way.
--
-- TODO: Don't cheat with fromList.
--
-- ((1,3),(2,4))
--
transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
-transpose m = Mat $ V.fromList column_list
+transpose matrix =
+ construct lambda
where
- column_list = [ column m i | i <- [0..(ncols m)-1] ]
+ lambda i j = matrix !!! (j,i)
-- | Is @m@ symmetric?
-- >>> fixed_point g eps u0
-- ((1.0728549599342185),(1.0820591495686167))
--
-vec1d :: (a) -> Mat N1 N1 a
+vec1d :: (a) -> Col1 a
vec1d (x) = Mat (mk1 (mk1 x))
-vec2d :: (a,a) -> Mat N2 N1 a
+vec2d :: (a,a) -> Col2 a
vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
-vec3d :: (a,a,a) -> Mat N3 N1 a
+vec3d :: (a,a,a) -> Col3 a
vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
-vec4d :: (a,a,a,a) -> Mat N4 N1 a
+vec4d :: (a,a,a,a) -> Col4 a
vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
-vec5d :: (a,a,a,a,a) -> Mat N5 N1 a
+vec5d :: (a,a,a,a,a) -> Col5 a
vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z))
-- Since we commandeered multiplication, we need to create 1x1
-- matrices in order to multiply things.
-scalar :: a -> Mat N1 N1 a
+scalar :: a -> Mat1 a
scalar x = Mat (mk1 (mk1 x))
dot :: (RealRing.C a, n ~ N1, m ~ S t, Arity t)
-- >>> diagonal m
-- ((1),(5),(9))
--
-diagonal :: (Arity m) => Mat m m a -> Mat m N1 a
+diagonal :: (Arity m) => Mat m m a -> Col m a
diagonal matrix =
construct lambda
where
let (Mat rows) = diagonal matrix
in
element_sum $ V.map V.head rows
+
+
+-- | Zip together two column matrices.
+--
+-- Examples:
+--
+-- >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
+-- >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
+-- >>> colzip m1 m2
+-- (((1,1)),((1,2)),((1,3)))
+--
+colzip :: Arity m => Col m a -> Col m a -> Col m (a,a)
+colzip c1 c2 =
+ construct lambda
+ where
+ lambda i j = (c1 !!! (i,j), c2 !!! (i,j))
+
+
+-- | Zip together two column matrices using the supplied function.
+--
+-- Examples:
+--
+-- >>> let c1 = fromList [[1],[2],[3]] :: Col3 Integer
+-- >>> let c2 = fromList [[4],[5],[6]] :: Col3 Integer
+-- >>> colzipwith (^) c1 c2
+-- ((1),(32),(729))
+--
+colzipwith :: Arity m
+ => (a -> a -> b)
+ -> Col m a
+ -> Col m a
+ -> Col m b
+colzipwith f c1 c2 =
+ construct lambda
+ where
+ lambda i j = f (c1 !!! (i,j)) (c2 !!! (i,j))
+
+
+-- | Map a function over a matrix of any dimensions.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> map2 (^2) m
+-- ((1,4),(9,16))
+--
+map2 :: (a -> b) -> Mat m n a -> Mat m n b
+map2 f (Mat rows) =
+ Mat $ V.map g rows
+ where
+ g = V.map f
+
+
+-- | Fold over the entire matrix passing the coordinates @i@ and @j@
+-- (of the row/column) to the accumulation function.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
+-- 18
+--
+ifoldl2 :: forall a b m n.
+ (Int -> Int -> b -> a -> b)
+ -> b
+ -> Mat m n a
+ -> b
+ifoldl2 f initial (Mat rows) =
+ V.ifoldl row_function initial rows
+ where
+ -- | The order that we need this in (so that @g idx@ makes sense)
+ -- is a little funny. So that we don't need to pass weird
+ -- functions into ifoldl2, we swap the second and third
+ -- arguments of @f@ calling the result @g@.
+ g :: Int -> b -> Int -> a -> b
+ g w x y = f w y x
+
+ row_function :: b -> Int -> Vec n a -> b
+ row_function rowinit idx r = V.ifoldl (g idx) rowinit r
+
+
projects / numerical-analysis.git / blobdiff